The condition (REF ) on the time step size implies \({\tau }=\mathcal {O}(h^2/\varepsilon ^2)\) ,
which is the same as those enforced in [1]}, [2]}.
This restriction comes essentially from the explicit term \(\Delta _hu^n\) 
due to the use of the Crank–Nicolson approximation,
which also means that
the second-order scheme (REF ) cannot preserve the MBP unconditionally
even though we introduce the stabilization term.
In practical computations, \( g(\widehat{u}^{n+\frac{1}{2}},\widehat{s}^{n+\frac{1}{2}})\approx 1\) 
so that the requirement for the time step size can be set to be \({\tau }\le ( \frac{\kappa }{2} + \frac{\varepsilon ^2}{h^2} )^{-1}\)  in order to preserve the MBP,
which is later used in our numerical experiments.
